A preliminary formalization of scale in topology optimization

Topology Optimization often enforces a minimum feature size, also known as a scale, to avoid instability and inaccuracy in the optimized result. This is usually done by applying a discretized convolution kernel on either the density or the sensitivity values. It is not well-established, however, exactly how this filtering approach enforces scale numerically, nor is it evident that the enforced scale is consistent. This paper introduces Linear Scale-Space Theory from the field of computer vision as a rigorous explanation for how a generic convolution enforces scale upon an optimization algorithm. This framework gives several guiding axioms for how to construct a filtering procedure in order to generate a consistent scale parameter across the entire problem domain. When this theory is reapplied to linear filtering in Topology Optimization, it is shown that the standard filtering approach results in inconsistent enforcement of scale on the boundaries of a volume. This formulation error must be corrected by expansion of the boundaries to include virtual elements which have a finite sensitivity but can accept no volume, also known as passive elements. Additionally, it is shown that irregular meshes result in inconsistent scale across the volume, which should be corrected by local modification of the convolution kernel. Numerical tests are performed for basic problems with expanded boundaries in both 2D and 3D. Significant changes in solution topology for these basic problems are noted and discussed. From this preliminary set of results, further avenues of research are suggested.

Name thesis:degree_name

M.S.

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Civil Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2018

Author dc:creator

• Vaught, Louis Oren

Contributors dc:contributor

• Elbanna, Ahmed

• Topology
• Optimization
• Structural
• Scale
• Scale-space

Statement dc:rights

• Copyright 2017 Louis Vaught

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/99425

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/99425